Qiiadrature of the Spherical Ellipse. 29 



digression from the main subject of this paper, that if a series 

 of concentric ellipses be described having coincident axes, and 

 the sum of their semiaxes constant equal to L suppose, the locus 

 of their points o^ linear section will be a hypocycloid, concentric 

 with the ellipses ; the radius of whose generating circle = L, 

 and the radius of whose rolling circle is = |^ L; and also that 

 the difference between the elliptic arcs is to the difference be- 

 tween the corresponding hypocycloidal arcs in the constant 

 ratio of 2 : 3. 



XXIII. A formula analogous to (31.) may be established 

 for the rectification of any curve on the surface of a sphere 

 formed by the intersection of this surface with a concentric 

 cone of any order. 



In the first place, let the cone be of the second degree, and 

 let a plane be drawn perpendicular to the axis of this cone, 

 touching the sphere and cutting the cone in the elliptic base; 

 let a tangent plane (T) be drawn to the cone, cutting the plane 

 of the elliptic base in a right line w, a tangent to this ellipse, 

 and the surface of the sphere in an arc of a great circle, touch- 

 ing the spherical ellipse ; let the distance from the centre of 

 the sphere to the point of contact of the tangent with the 

 ellipse be R; through the centre of the sphere let a plane (H) 

 be drawn perpendicular to u, then as m is a right line as well 

 in the plane (T) as in the elliptic base, the plane (H) is per- 

 pendicular both to the tangent plane (T) and to the base of 

 the cone; hence the plane (H) passes through the axis of the 

 cone and the centre of the plane ellipse, as also of the spheri- 

 cal ellipse, cutting the former in a perpendicular p from the 

 centre on the tangent u, and the latter in an arc cr of a great 

 circle, perpendicular to the tangent arc to the spherical ellipse; 

 for the two latter arcs must be at right angles to each other, 

 since the planes (T) and (H) are at right angles. Let P be 

 the distance from the centre of the sphere to the point where 

 the plane (H) cuts the right line w, r the distance from the 

 centre of the plane ellipse to the point of contact of « with it; 

 then to any one attending to this construction it will be mani- 

 fest that (c being the radius of the sphere) 



R2 = c2 + r% P2 = c^ + p% R2 = P2 + u^ . (45.) 



XXIV. Let rf 5 be the element of an arc of the ellipse be- 

 tween any two consecutive values of R indefinitely near to each 

 other, cd<r the corresponding element of the spherical ellipse 

 between the same consecutive positions of R ; then the areas 

 of the elementary triangles on the surface of the cone between 

 these consecutive positions of R having their vertices at the 

 centre of the sphere and their bases an element of the arc of 

 the ellipse and of an arc of the spherical ellipse respectively, 



